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In category theory, a branch of mathematics, a section is a right inverse of some morphism. Dually, a retraction is a left inverse of some morphism. In other words, if ''f'' : ''X'' → ''Y'' and ''g'' : ''Y'' → ''X'' are morphisms whose composition ''f'' o ''g'' : ''Y'' → ''Y'' is the identity morphism on ''Y'', then ''g'' is a section of ''f'', and ''f'' is a retraction of ''g''. Every section is a monomorphism, and every retraction is an epimorphism. In algebra the sections are also called split monomorphisms and the retractions split epimorphisms. In an abelian category, if ''f'' : ''X'' → ''Y'' is a split epimorphism with split monomorphism ''g'' : ''Y'' → ''X'', then ''X'' is isomorphic to the direct sum of ''Y'' and the kernel of ''f''. ==Examples== In the category of sets, every monomorphism (injective function) with a non-empty domain is a section and every epimorphism (surjective function) is a retraction; the latter statement is equivalent to the axiom of choice. In the category of vector spaces over a field ''K'', every monomorphism and every epimorphism splits; this follows from the fact that linear maps can be uniquely defined by specifying their values on a basis. In the category of abelian groups, the epimorphism Z→Z/2Z which sends every integer to its image modulo 2 does not split; in fact the only morphism Z/2Z→Z is the 0 map. Similarly, the natural monomorphism Z/2Z→Z/4Z doesn't split even though there is a non-trivial homomorphism Z/4Z→Z/2Z. The categorical concept of a section is important in homological algebra, and is also closely related to the notion of a section of a fiber bundle in topology: in the latter case, a section of a fiber bundle is a section of the bundle projection map of the fiber bundle. Given a quotient space with quotient map , a section of is called a transversal. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Section (category theory)」の詳細全文を読む スポンサード リンク
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